In order to find the conformal factor relating \({\mathbf {\ell }}\)  and \(\bar{{\mathbf {q}}}\) 
we follow the same procedure used to show that the (4-dimensional) de Sitter spacetime
can be conformally embedded in the (4-dimensional) Einstein static
Universe —see e.g. Section 6.3 in [1]}. More precisely,
starting from the metric \({\mathbf {\ell }}\)  of the unit hyperboloid, introduce
the coordinate transformation given by the relation
\(\mathbf {d}\chi = \cosh \chi \mathbf {d}\tau .\) 
